The set of numbers of the form

a + b \sqrt{2}

where

a

and

b

are integers form a mathematical structure called a ring. It is easy to show that

R

is closed for addition, that 0 belongs to

R

and is the additive identity and that every number in

R

has an additive inverse which is in

R

. Also addition of numbers in

R

is associative so this is an additive group.

What about multiplication?

Again it is easy show that $R$ is closed for multiplication, that 1 belongs to $R$ and is the multiplicative identity and that multiplication of numbers in $R$ is associative. However it is also easy to find a counter example to show that not every number in $R$ has a multiplicative inverse which is in $R$
(Try this for youself, for example look for an inverse for $(2 + 3 \sqrt{2})$ and you will find that it would have to be $(\frac{- 1}{7} + \frac{3 \sqrt{2}}{14})$ but this number is not in $R$ . NB. $R$ is the set of numbers $a + b \sqrt{2}$ where $a$ and $b$ are integers).

So we can add, subtract and multiply these numbers. If $u, v$ and $w$ are in the set $R$ it is easy to show that the distributive property holds: $u (v + w) = uv + uw$ . So we have some of the same structure as the arithmetic of real numbers but without division. This structure is called a ring.